When can two matrices be added or subtracted?

When they have the same order; the operation is done entry by entry.

When is the product \mathbf{AB} defined?

When the number of columns of \mathbf{A} equals the number of rows of \mathbf{B}.

How is entry (i,j) of \mathbf{AB} found?

Multiply row i of \mathbf{A} by column j of \mathbf{B}, entry by entry, and add.

Is matrix multiplication commutative?

No: in general \mathbf{AB}\ne\mathbf{BA}.

Is matrix multiplication associative?

Yes: (\mathbf{AB})\mathbf{C}=\mathbf{A}(\mathbf{BC}).

What is the zero matrix \mathbf{O}?

The matrix with every entry 0; \mathbf{A}+\mathbf{O}=\mathbf{A}.

What is the 2\times2 identity matrix?

\begin{pmatrix}1&0\\0&1\end{pmatrix}.

What does \mathbf{AI} equal for a square matrix \mathbf{A}?

\mathbf{A} (and \mathbf{IA}=\mathbf{A}).

If \mathbf{AB}=\mathbf{O}, must \mathbf{A} or \mathbf{B} be \mathbf{O}?

No: two non-zero matrices can multiply to give \mathbf{O}.

How do you write 3 in a 2\times2 matrix equation?

As 3\mathbf{I}=\begin{pmatrix}3&0\\0&3\end{pmatrix}.

Matrix arithmeticAQA A-Level Further Maths: Flashcards

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What is the order of a matrix with $m$ rows and $n$ columns?

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What is the order of a matrix with $m$ rows and $n$ columns?: $m\times n$ .
When can two matrices be added or subtracted?: When they have the same order; the operation is done entry by entry.
What does multiplying a matrix by a scalar $k$ do?: Multiplies every entry by $k$ .
When is the product $\mathbf{AB}$ defined?: When the number of columns of $\mathbf{A}$ equals the number of rows of $\mathbf{B}$ .
Order of $\mathbf{AB}$ if $\mathbf{A}$ is $m\times n$ and $\mathbf{B}$ is $n\times p$ ?: $m\times p$ .
How is entry $(i,j)$ of $\mathbf{AB}$ found?: Multiply row $i$ of $\mathbf{A}$ by column $j$ of $\mathbf{B}$ , entry by entry, and add.
Is matrix multiplication commutative?: No: in general $\mathbf{AB}\ne\mathbf{BA}$ .
Is matrix multiplication associative?: Yes: $(\mathbf{AB})\mathbf{C}=\mathbf{A}(\mathbf{BC})$ .
What is the zero matrix $\mathbf{O}$ ?: The matrix with every entry $0$ ; $\mathbf{A}+\mathbf{O}=\mathbf{A}$ .
What is the $2\times2$ identity matrix?: $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ .
What does $\mathbf{AI}$ equal for a square matrix $\mathbf{A}$ ?: $\mathbf{A}$ (and $\mathbf{IA}=\mathbf{A}$ ).
If $\mathbf{AB}=\mathbf{O}$ , must $\mathbf{A}$ or $\mathbf{B}$ be $\mathbf{O}$ ?: No: two non-zero matrices can multiply to give $\mathbf{O}$ .
How do you write $3$ in a $2\times2$ matrix equation?: As $3\mathbf{I}=\begin{pmatrix}3&0\\0&3\end{pmatrix}$ .

Exam questions on Matrix arithmetic

$\mathbf{C}=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $\mathbf{D}=\begin{pmatrix}0&-1\\5&2\end{pmatrix}$ .
Given that $\mathbf{C}+\mathbf{X}=3\mathbf{I}$ , where $\mathbf{I}$ is the $2\times2$ identity matrix, find $\mathbf{X}$ .2 marks
$\mathbf{P}=\begin{pmatrix}1&0&2\\-1&3&1\end{pmatrix}$ and $\mathbf{Q}=\begin{pmatrix}2&1\\0&-1\\4&3\end{pmatrix}$ .
Find $\mathbf{PQ}$ .2 marks
$\mathbf{M}=\begin{pmatrix}2&-1\\1&0\end{pmatrix}$ .
Find $\mathbf{M}^2$ and show that $\mathbf{M}^2=2\mathbf{M}-\mathbf{I}$ , where $\mathbf{I}$ is the $2\times2$ identity matrix.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).